Update app.py

app.py

@@ -528,6 +528,7 @@ STRICT REQUIREMENTS:
    - Put each solution step on its own line in $$ $$
    - DO NOT use \\begin{{aligned}} or similar environments
 5. Include a detailed solution
+   - Begin the solution with a heading in bold and larger font size: Original Solution
    - If the question involves geometry make sure to identify any general geometric formulas that apply, For example:
         * Areas/volumes of common shapes and solids
         * Cross-sectional areas of geometric figures
@@ -536,7 +537,8 @@ STRICT REQUIREMENTS:
         * carefully consider the direction of change in variables
         * ensure integration bounds align with the physical direction of the process being modeled
 6. Maintain clear formatting
-7. At the end of the LaTeX solution output, print SymPy code that you would use to solve or verify the main equations in the question.
+7. At the end of the solution output, print SymPy code that you would use to solve or verify the main equations in the question 
+   and give this section a heading in bold and larger font size: SymPy Verification
 8. Observe the folloiwng SymPy Guidelines
 {SYMPY_GUIDELINES}"""
 
@@ -578,7 +580,7 @@ STRICT REQUIREMENTS:
                 response_text = f"{response_text}\n\nSymPy Verification Results:\n```\n{sympy_output}\n```\n\nVerification Analysis:\n{resolution_text}"
                 
                 # For difficulty level 5 AND when there's a discrepancy with a revised solution
-                if difficulty == 5 and has_discrepancy and revised_solution:
+                if has_discrepancy and revised_solution:
                     logger.debug("Performing final verification for difficulty 5 problem with discrepancy")
                     final_verification = perform_final_verification(revised_solution)
                     response_text += "\n\nFinal Expert Verification:\n" + final_verification
@@ -664,6 +666,8 @@ def check_and_resolve_discrepancy(initial_response, sympy_output):
         resolution_prompt = f"""Here is a mathematics question with two answers.
         The first, called Original solution, is a complete solution.
         The second, called SymPy Verification, will only provide the final answer.
+
+        Begin this section with a heading in bold and larger font size: Comparison of SymPy Verifiation with Original Solution
         
         If the SymPy Verification answer is consistent with the final answer Original solution,
         then please say that they are consistent and briefly explain why. 
@@ -674,7 +678,7 @@ def check_and_resolve_discrepancy(initial_response, sympy_output):
         3. Write "Here is the revised complete solution:" and then write out the ENTIRE solution from beginning 
            to end, including all parts that were correct and the corrections for any incorrect parts. 
            Do not refer to the original solution or say things like "the rest remains the same" - write 
-           out everything in full.
+           out everything in full. Begin this section with a heading in bold and larger font size: Revised Complete Solution
 
 Original solution:
 {initial_response}
@@ -798,6 +802,7 @@ Your complete solution must:
 - Maintain proper LaTeX formatting with $ for inline math and $ on separate lines
 - Include any relevant SymPy code for verification
 
+Begin this section with a heading in bold and larger font size: Final Verified Solution
 Remember to write out the complete solution even if you only need to add a few steps - the goal is to have a single, complete, verified solution that includes everything necessary for full mathematical rigor."""
 
     try: